All of the 4th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$2.50$ each for students, and the group paid $$32.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$12.00$ each for teachers and $$8.50$ each for students, and the group paid $$92.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+2.5y = 32}$ ${12x+8.5y = 92}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-12x-5y = -64}$ ${12x+8.5y = 92}$ Add the top and bottom equations together. $ 3.5y = 28 $ $ y = \dfrac{28}{3.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {6x+2.5y = 32}$ to find $x$ ${6x + 2.5}{(8)}{= 32}$ $6x+20 = 32$ $6x = 12$ $x = \dfrac{12}{6}$ ${x = 2}$ You can also plug ${y = 8}$ into $ {12x+8.5y = 92}$ and get the same answer for $x$ ${12x + 8.5}{(8)}{= 92}$ ${x = 2}$ There were $2$ teachers and $8$ students on the field trips.